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The visitors, coming off their first win at Manchester United for 21 years on Wednesday, controlled the first 40 minutes with 68% possession before Arsenal had chances in the final five minutes of the half.
Despite seeing plenty of the ball, Everton failed to get any of their five shots on target, with Kevin Mirallas blazing the clearest opening over the top.
Everton goalkeeper Tim Howard did not have a save to make until the 42nd minute but was then twice alert to keep the score level.
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